HST 583 fMRI DATA ANALYSIS AND ACQUISITION A Review of Statistics for fMRI Data Analysis Emery N. Brown Massachusetts General Hospital Harvard Medical School/MIT Division of Health, Sciences and Technology December 2, 2002

Outline What Makes Up an fMRI Signal? Statistical Modeling of an fMRI Signal Maxmimum Likelihoood Estimation for fMRI Data Analysis Conclusions

THE STATISTICAL PARADIGM (Box, Tukey) Question Preliminary Data (Exploration Data Analysis) Models Experiment (Confirmatory Analysis) Model Fit Goodness-of-Fit not satisfactory Assessment Satisfactory Make an Inference Make a Decision

Case 3: fMRI Data Analysis Question: Can we construct an accurate statistical model to describe the spatial temporal patterns of activation in fMRI images from visual and motor cortices during combined motor and visual tasks? (Purdon et al., 2001; Solo et al., 2001) A STIMULUS-RESPONSE EXPERIMENT Acknowledgements: Chris Long and Brenda Marshall

What Makes Up An fMRI Signal? Hemodynamic Response/MR Physics             i) stimulus paradigm a) event-related b) block ii) blood flow iii) blood volume iv) hemoglobin and deoxy hemoglobin content Noise Stochastic i) physiologic ii) scanner noise Systematic i) motion artifact ii) drift iii) [distortion] iv) [registration], [susceptibility]

Physiologic Response Model: Block Design

Gamma Hemodynamic Response Model

Physiologic Model: Event-Related Design

Physiologic Model: Flow, Volume and Interaction Terms

Scanner and Physiologic Noise Models

DATA: The sequence of image intensity measurements on a single pixel.

fMRI Signal and Noise Model Measurement on a single pixel at time Physiologic response Activation coefficient Physiologic and Scanner Noise for We assume the are independent, identically distributed Gaussian random variables.

fMRI Signal Model Physiologic Response hemodynamic response input stimulus Gamma model of the hemodynamic response Assume we know the parameters of g(t).

MAXIMUM LIKELIHOOD Define the likelihood function , the joint probability density viewed as a function of the parameter with the data fixed. The maximum likelihood estimate of is That is, is a parameter value for which attains a maximum as a function of for fixed

ESTIMATION Joint Distribution Log Likelihood Maximum Likelihood

GOODNESS-OF-FIT/MODEL SELECTION An essential step, if not the most essential step in a data analysis, is to measures how well the model describes the data. This should be assessed before the model is used to make inferences about that data. Akaike’s Information Criterion For maximum likelihood estimates it measures the trade-off between maximizing the likelihood (minimizing ) and the numbers of parameters the model requires.

GOODNESS-OF-FIT Residual Plots: KS Plots: We can check the Gaussian assumption with our K-S plots. Measure correlation in the residuals to assess independence.

EVALUATION OF ESTIMATORS Given an estimator of based on Mean-Squared Error: Bias= Consistency: Efficiency: Achieves a minimum variance (Cramer-Rao Lower Bound)

FACTOIDS ABOUT MAXIMUM LIKELIHOOD ESTIMATES Generally biased. Consistent, hence asymptotically unbiased. Asymptotically efficient. Variance can be approximated by minus the inverse of the Fisher information matrix. If is the estimate of then is the estimate of

Cramer-Rao Lower Bound CRLB gives the lowest bound on the variance of an estimate.

CONFIDENCE INTERVALS The approximate probability density of the maximum likelihood estimates is the Gaussian probability density with mean and variance where is the Fisher information matrix An approximate confidence interval for a component of is

THE INFORMATION MATRIX

CONFIDENCE INTERVAL

Kolmogorov-Smirnov Test White Noise Model

Pixelwise Confidence Intervals for the Slice White Noise Model Pixelwise Confidence Intervals for the Slice

fMRI Signal and Noise Model 2 Measurement on a single pixel at time Physiologic response Activation coefficient Physiologic and Scanner Noise for We assume the are correlated noise AR(1) Gaussian random variables.

Simple Convolution Plus Correlated Noise

Kolmogorov-Smirnov Test Correlated Noise Model

Pixelwise Confidence Intervals for the Slice Correlated Noise Model Pixelwise Confidence Intervals for the Slice

AIC Difference = AIC Colored Noise-AIC White Noise

fMRI Signal and Noise Model 3 Measurement on a single pixel at time Physiologic response Physiologic and Scanner Noise for We assume the are independent, identically distributed Gaussian random variables.

Harmonic Regression Plus White Noise Model

AIC Difference Map= AIC Correlated Noise-AIC Harmonic Regression

Conclusions The white noise model gives a good description of the hemodynamic response The correlated noise model incorporates known physiologic and biophysical properties and hence yields a better fit The likelihood approach offers a unified way to formulate a model, compute confidence intervals, measure goodness of fit and most importantly make inferences.